Let G be a group and A and B be normal subgroups of G such that A is isomorphic to B. Show by an example that G/A is NOT isomorphic G/B.

Question

I let G = $(\mathbb{Z},+)$ and let $A = 2\mathbb{Z}$ and $B = 3\mathbb{Z}$...which I thought would work.

I can show these are are normal subgroups and that they are isomorphic easily.

However, I am having trouble showing that $G/2\mathbb{Z}$ is not isomorphic to $G/3\mathbb{Z}$.

Help is appreciated! If there is a better G, A, B to use here, suggestions are also welcome.

Answer 1

An isomorphism is a bijection, however $|G/A|=2$ and $|G/B|=3$.